Theorem isstruct2 17109

Description: The property of being a structure with components in (1^st ‘𝑋)...(2^nd ‘𝑋). (Contributed by Mario Carneiro, 29-Aug-2015.)

Assertion

Ref	Expression
isstruct2	⊢ (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Proof of Theorem isstruct2

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brstruct 17108	. . 3 ⊢ Rel Struct
2	1	brrelex12i 5727	. 2 ⊢ (𝐹 Struct 𝑋 → (𝐹 ∈ V ∧ 𝑋 ∈ V))
3		ssun1 4168	. . . . 5 ⊢ 𝐹 ⊆ (𝐹 ∪ {∅})
4		undif1 4471	. . . . 5 ⊢ ((𝐹 ∖ {∅}) ∪ {∅}) = (𝐹 ∪ {∅})
5	3, 4	sseqtrri 4015	. . . 4 ⊢ 𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅})
6		simp2 1135	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → Fun (𝐹 ∖ {∅}))
7	6	funfnd 6578	. . . . . 6 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}))
8		elinel2 4192	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ (ℕ × ℕ))
9		1st2nd2 8026	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℕ × ℕ) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
10	8, 9	syl 17	. . . . . . . . . 10 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
11	10	3ad2ant1 1131	. . . . . . . . 9 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
12	11	fveq2d 6895	. . . . . . . 8 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) = (...‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩))
13		df-ov 7417	. . . . . . . . 9 ⊢ ((1st ‘𝑋)...(2nd ‘𝑋)) = (...‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
14		fzfi 13961	. . . . . . . . 9 ⊢ ((1st ‘𝑋)...(2nd ‘𝑋)) ∈ Fin
15	13, 14	eqeltrri 2825	. . . . . . . 8 ⊢ (...‘⟨(1st ‘𝑋), (2nd ‘𝑋)⟩) ∈ Fin
16	12, 15	eqeltrdi 2836	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (...‘𝑋) ∈ Fin)
17		difss 4127	. . . . . . . . 9 ⊢ (𝐹 ∖ {∅}) ⊆ 𝐹
18		dmss 5899	. . . . . . . . 9 ⊢ ((𝐹 ∖ {∅}) ⊆ 𝐹 → dom (𝐹 ∖ {∅}) ⊆ dom 𝐹)
19	17, 18	ax-mp 5	. . . . . . . 8 ⊢ dom (𝐹 ∖ {∅}) ⊆ dom 𝐹
20		simp3 1136	. . . . . . . 8 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom 𝐹 ⊆ (...‘𝑋))
21	19, 20	sstrid 3989	. . . . . . 7 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ⊆ (...‘𝑋))
22	16, 21	ssfid 9283	. . . . . 6 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → dom (𝐹 ∖ {∅}) ∈ Fin)
23		fnfi 9197	. . . . . 6 ⊢ (((𝐹 ∖ {∅}) Fn dom (𝐹 ∖ {∅}) ∧ dom (𝐹 ∖ {∅}) ∈ Fin) → (𝐹 ∖ {∅}) ∈ Fin)
24	7, 22, 23	syl2anc 583	. . . . 5 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∖ {∅}) ∈ Fin)
25		p0ex 5378	. . . . 5 ⊢ {∅} ∈ V
26		unexg 7745	. . . . 5 ⊢ (((𝐹 ∖ {∅}) ∈ Fin ∧ {∅} ∈ V) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
27	24, 25, 26	sylancl 585	. . . 4 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V)
28		ssexg 5317	. . . 4 ⊢ ((𝐹 ⊆ ((𝐹 ∖ {∅}) ∪ {∅}) ∧ ((𝐹 ∖ {∅}) ∪ {∅}) ∈ V) → 𝐹 ∈ V)
29	5, 27, 28	sylancr 586	. . 3 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝐹 ∈ V)
30		elex 3488	. . . 4 ⊢ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) → 𝑋 ∈ V)
31	30	3ad2ant1 1131	. . 3 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → 𝑋 ∈ V)
32	29, 31	jca 511	. 2 ⊢ ((𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)) → (𝐹 ∈ V ∧ 𝑋 ∈ V))
33		simpr 484	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
34	33	eleq1d 2813	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ↔ 𝑋 ∈ ( ≤ ∩ (ℕ × ℕ))))
35		simpl 482	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → 𝑓 = 𝐹)
36	35	difeq1d 4117	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (𝑓 ∖ {∅}) = (𝐹 ∖ {∅}))
37	36	funeqd 6569	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (Fun (𝑓 ∖ {∅}) ↔ Fun (𝐹 ∖ {∅})))
38	35	dmeqd 5902	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → dom 𝑓 = dom 𝐹)
39	33	fveq2d 6895	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (...‘𝑥) = (...‘𝑋))
40	38, 39	sseq12d 4011	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → (dom 𝑓 ⊆ (...‘𝑥) ↔ dom 𝐹 ⊆ (...‘𝑋)))
41	34, 37, 40	3anbi123d 1433	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑥 = 𝑋) → ((𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥)) ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
42		df-struct 17107	. . 3 ⊢ Struct = {⟨𝑓, 𝑥⟩ ∣ (𝑥 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝑓 ∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))}
43	41, 42	brabga 5530	. 2 ⊢ ((𝐹 ∈ V ∧ 𝑋 ∈ V) → (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋))))
44	2, 32, 43	pm5.21nii 378	1 ⊢ (𝐹 Struct 𝑋 ↔ (𝑋 ∈ ( ≤ ∩ (ℕ × ℕ)) ∧ Fun (𝐹 ∖ {∅}) ∧ dom 𝐹 ⊆ (...‘𝑋)))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ∖ cdif 3941   ∪ cun 3942   ∩ cin 3943   ⊆ wss 3944  ∅c0 4318  {csn 4624  ⟨cop 4630   class class class wbr 5142   × cxp 5670  dom cdm 5672  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986  Fincfn 8955   ≤ cle 11271  ℕcn 12234  ...cfz 13508   Struct cstr 17106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-struct 17107

This theorem is referenced by:  structn0fun  17111  isstruct  17112  setsstruct2  17134